The reigning champions ultimately ran out comfortable 3-0 winners on Saturday to continue their perfect start to life under Allegri.
However, things could have been significantly different as Giorgio Chiellini conceded a 59th-minute penalty, with Juve 1-0 up at the time courtesy of Carlos Tevez's opener.
Buffon threw himself down to his left to palm away German Denis' spot-kick and Tevez doubled Juve's lead seconds later.
Alvaro Morata eventually wrapped up the win in the final 10 minutes, securing victory with his first goal for the club.
Allegri was left hailing the importance of Buffon.
He told Sky Sport Italia: "It was a great victory, Roma won earlier, so there was even more pressure on us to win. It wasn't easy, as Atalanta are particularly strong on home turf.
"We struggled in the opening 10 minutes and didn't clear the ball properly, so we tended to be caught out. The lads did well, then Buffon was decisive for the result.
"I think the difference between Buffon and all other goalkeepers is that he can barely be tested throughout the game, yet still come up with the decisive save when he's needed.
"That’s how you can tell a real champion."
Tevez echoed Allegri's comments about Buffon and acknowledged the importance of striking Atalanta while they were already down.
"We all know Buffon is great, the whole world knows it," he said.
"After that we had to react and score straight away, which thankfully we did."
